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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeOct 8th 2010

    I added some simpler motivation in terms of the basic example to the beginning of distributive law.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 11th 2010
    Thanks, nice.
    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 12th 2010

    Yes, nice. Rings make an obvious choice to illustrate a distributive law between monads. Then we’re introduced to the scary thought that there’s a whole 2-category of such laws. Could this be illustrated by a surprising example or two of a distributive law, perhaps something one would never have thought to be such a law?

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 12th 2010

    one would never have thought to be such a law

    The level of difficulty in fullfilling this task depends on who is "one" who never thought...

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeOct 12th 2010

    There’s definitely a gap between the first two sections of this page which someone who has some time on their hands should fill. (-:

    • CommentRowNumber6.
    • CommentAuthorFinnLawler
    • CommentTimeOct 12th 2010

    a surprising example

    I for one was quite tickled to find out that factorization systems are distributive laws between categories taken as monads in Span(Set). That might be a good non-obvious example.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 13th 2010

    Good. I’ve put that example in.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2010

    It’s not quite true as stated, though – ordinary distributive laws in Span(Set) only give you “strict” factorization systems. I corrected it.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2012

    I keep pointing to “distributive law” whenever I mention something distributive. Now looking back at the entry to see what it actually says, I notice that many of the common “distributivities” were not really mentioned. I added a quick pointer to distributive category and to tensor products distributing over direct sums. But this needs to be better incorporated into the entry, eventually.

    • CommentRowNumber10.
    • CommentAuthorPaoloPerrone
    • CommentTimeSep 28th 2019

    Added explicit diagrams.

    diff, v26, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2022

    added these pointers:

    diff, v30, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2022
    • (edited Dec 16th 2022)

    added cross-link to the new entry weak distributive law and to pseudo-distributive law

    diff, v36, current

    • CommentRowNumber13.
    • CommentAuthorSam Staton
    • CommentTimeJul 6th 2023

    mention monad lifting characterization

    diff, v37, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2023

    spelled out (here) the mixed distributivity law of a monad over a comonad (from Brookes & Van Stone 1993 Def. 3)

    diff, v41, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2023

    added (here) also statement of the resulting two-sided Kleisli composition

    diff, v42, current

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeOct 1st 2023

    distributivity law of a monad over a comonad

    I would just like to warn that the prepositions etc. in phrases for distributive laws are not standardized. One hear law from monad to comonad, from comonad to monad, between monad and comonad, between comonad and monad etc. and I never found a way to logically remember any of those. I mean for the mixed law (entwining structure) there is no problem – there is just one, but when there are two monads or two comonads, there are two different laws – then it does matter which one is first and which the second, but which order corresponds to whose usage of grammar is totally random.

    • CommentRowNumber17.
    • CommentAuthorvarkor
    • CommentTimeOct 1st 2023

    I would just like to warn that the prepositions etc. in phrases for distributive laws are not standardized.

    There is (what ought to be) a standard, but I have heard people get it wrong. The phrasing should align with “multiplication distributes over addition”, which means that we have a transformation ×++×{\times} \circ {+} \Rightarrow {+} \circ {\times}. I.e. SS distributes over TT when we have a transformation STTSST \Rightarrow TS. Using any other terminology or convention is at best ambiguous and at worst misleading.

    I notice the nLab page is not entirely consistent about this and talks about “distributives law from one thing to another”. If no-one has any objections, I’ll clean this up.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2023

    In the mixed case I have been following Brookes & Van Stone 1993, Def. 3 who speak the other way around p. 12.

    But it’s opposite to what one should say, yes.

    Incidentally, Brookes & Van Stone is the only reference for the mixed case that I am aware of (in my ignorance). Maybe we can find another reference for the mixed case which has the terminology straight.

    • CommentRowNumber19.
    • CommentAuthorvarkor
    • CommentTimeOct 1st 2023

    Maybe we can find another reference for the mixed case which has the terminology straight.

    Power and Wanatabe use the expected terminology (opposite to that of Brookes and Van Stone) in Combining a monad and a comonad (e.g. see the introduction, or Definition 6.1 on p. 153).

    • CommentRowNumber20.
    • CommentAuthorzskoda
    • CommentTimeOct 1st 2023
    • (edited Oct 1st 2023)

    Urs 18, I repeat – in the case of a monad and comonad there are NO two versions just one, so the direction matters only when there are two monads or two comonads.

    The notion is also self-dual in the case of a monad and comonad, as Brzezinski put it, if you reverse all the arrows (then monads exachange for comonads, algebras with coalgebras etc.) you get the same diagram.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2023

    varkor, thanks,

    I have adjusted the wording here and added pointer to Power & Watanabe.

    Unfortunately, Power & Watanabe don’t seem to explicitly state the distributivity of a comonad over a monad(?) – their Def. 6.1 only states the distributivity of a monad over a comonad. Not that it’s a big deal, but it means one can’t quite cite them for it.

    Also, they attribute the notion of comonad distributing over a monad to Brookes & Geva 1991 (pdf, instead of Brookes & Van Stone) but I don’t spot any discussion of distributivity in there(?)

    diff, v44, current

    • CommentRowNumber22.
    • CommentAuthorzskoda
    • CommentTimeOct 1st 2023
    • (edited Oct 1st 2023)

    Mixed distributive laws are stated and used in 1970s also in

    • Donovan H. Van Osdol, Bicohomology theory, Trans. Amer. Math. Soc. 183 (1973), 449-476 doi

    Now I see that Brookes and Van Stone use inverse-looking map, but it is doubtful to call it a distributive law. I mean the following.

    Look, if you have two monads, TT, SS, then a distributive law TSSTTS \to ST lifts SS to a monad on C TC^T and the opposite law STTSST\to TS lifts TT to a monad on C SC^S. Both are useful in the lifting sense.

    If you have a monad and a comonad, say comonad GG and monad SS then SGGSSG\to GS gives simultaneously both lift of GG to a comonad on C SC^S and SS to a monad on C GC^G. Namely if MM is a GG-comodule you do composition SMSGMGSMSM\to SGM\to GSM to obtain a lifted GG-comodule. If MM is an SS-module you do a composition SGMGSMGMSGM\to GSM\to GM to get a lifted SS-module. In that basic sense, in both cases you need a distributive law SGGSSG\to GS, unlike the two monads case and unlike the two comonads case. This is the reason most references have only one distributive law for mixed case and people in “entwining structure” community repeat that there is only one. I do not understand how to parallel Brookes and Van Stone concept (apart from trivial inversion of the law in the commutative diagrams); how can it be interpreted via any lift ??

    P.S. bialgebras in the sense of van Osdol and entwining modules of Brzezinski-Majid are basically the same thing.

    P.S.2 Maybe if we interchange EM lifts to Kleisli lifts it would change to the other distributive law. This is again unlike the two monad or two comonads case, but is reasonable as it is 2-categorical setup so it is another duality op versus co.

    • CommentRowNumber23.
    • CommentAuthorzskoda
    • CommentTimeOct 1st 2023

    I am confused now. While the question of lifts above is easily checked, after my crosschecks within old literature (including a paper on entwinings of my own) I need some time to reconcile the two points of view. I remember that Gabi has sent some unexpected dual comonad picture for something similar but different from entwining modules in certain MathReview. I need to compare that and will be back later.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2023
    • (edited Oct 1st 2023)

    added a remark (here) on the issue of terminology

    (noting that already the original terminology in Beck 1969 p. 96 is “backwards”!)

    also changed all occurrences of “distributivity from … to …” (introduced in revision 1) to “distributivity of …. over …”

    diff, v45, current

    • CommentRowNumber25.
    • CommentAuthorzskoda
    • CommentTimeOct 1st 2023
    • (edited Oct 1st 2023)

    Wisbauer’s survey

    • R. Wisbauer, Algebras versus coalgebras, Appl. Categ. Structures 16 (2008), no. 1-2, 255–295 doi

    states a theorem 5.4 which says roughly what I argued above for the distributive law SGGSSG\to GS (citation from above)

    If you have a monad and a comonad, say comonad GG and monad SS then SGGSSG\to GS gives simultaneously both lift of GG to a comonad on C SC^S and SS to a monad on C GC^G. Namely if MM is a GG-comodule you do composition SMSGMGSMSM\to SGM\to GSM to obtain a lifted GG-comodule. If MM is an SS-module you do a composition SGMGSMGMSGM\to GSM\to GM to get a lifted SS-module. In that basic sense, in both cases you need a distributive law SGGSSG\to GS, unlike the two monads case and unlike the two comonads case.

    Street in

    uses the other version and notices that it is a mate to the distributive laws among two monads (edit: provided the comonad has a right adjoint).

    For algebra and a coalgebra one can look at algebra as defining a monad both to the category of left and the category of right modules (edit: over a ground ring). So, in my understanding, for a coalgebra CC and algebra AA, the law ψ:CAAC\psi: C\otimes A\to A\otimes C can pertain to either of the two kind of laws, as the order is different when we tensor multiply from the left and from the right.

    • CommentRowNumber26.
    • CommentAuthorzskoda
    • CommentTimeOct 1st 2023
    • (edited Oct 1st 2023)

    Yes, Power and Watanabe reconcile what was confusing me

    Observe that this distributive law allowing one to make a two-sided version of a Kleisli construction is in the opposite direction to that required to build a category of bialgebras.

    This explains/acknowledges why the people interested in bialgebras/entwining modules like Brzezinski and me, who look for lifting property and the composed coring see that all 3 constructions (two liftings and the composed coring) require the same direction of the distributive law. I was not aware of this statement about Kleisli construction. (In fact once I complained to Tomasz that the other mixed distributive law seemed to be useful for something and he warned me that there is anyway only one kind for all 3 motivating purposes.)

    This corrects my first line in 20 (which is still true if understood in the sense of lifting solution).

    • CommentRowNumber27.
    • CommentAuthorzskoda
    • CommentTimeOct 2nd 2023
    • Liang Ze Wong, Distributive laws, post at nn-cafe, Feb 2017

    diff, v47, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2023

    I have slightly re-arranged the (few) sentences under “Big picture” (here), for clarity. But this still deserves to be expanded on more.

    diff, v50, current

    • CommentRowNumber29.
    • CommentAuthormattecapu
    • CommentTimeJan 19th 2024

    added distributive law as monad in monads

    diff, v52, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2024
    • (edited Jan 19th 2024)

    I have added some Definition/Remark-environments to structure the text in this subsection (there is room to do more of this).

    Notice that our Instiki parser does not render code like

    $st \rightarrow ts$
    

    as expected, instead it requires whitespace for separating variable names:

    $s t \rightarrow t s$
    

    I have fixed this now.

    diff, v53, current

    • CommentRowNumber31.
    • CommentAuthorvarkor
    • CommentTimeFeb 11th 2024

    Added a reference showing that there may be an arbitrary number of distributive laws between two monads.

    diff, v54, current